Orientation Distribution Function for Diffusion MRI

Orientation Distribution Function for Diffusion MRI Evgeniya Balmashnova 28 October 2009

Diffusion Tensor Imaging

Diffusion MRI

Diffusion MRI P(r, t) = 1 (4πDt) 3/2 e 1 4t r 2 D 1 t Diffusion time D Diffusion coefficient P Probability of travel to point r in time t

Diffusion Tensor Imaging P(r, t) = 1 (4π D t) 3/2 e 1 4t rt D 1 r D = D xx D xy D xz D yx D yy D yz D zx D zy D zz

Diffusion Tensor Imaging: Application

Diffusion Tensor Imaging

MRI and Diffusion Tensor Imaging

High Angular Resolution Diffusion Imaging S(g) = S 0 e bd(g) (Stejskal & Tanner, 1965)

HARDI

Diffusion Orientation Distribution Function

Alternative Decompositions Spherical harmonics N l D N (g) = c lm Y lm (g) l=0 m= l High order tensors 3 3 D N (g) =... D i 1...i N g i1... g in i 1 =1 i N =1 Hierarchial tensors N 3 3 D N (g) =... D i 1...i l g i1... g il (Florack & Balmashnova, 2008) l=0 i 1 =1 i N =1

Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms

1. E 0 (D 0 ) = (S(g) D 0 ) 2 dω, Ω 2. E n(d i 1...in n 1 ) = ((S(g) D i 1...i k g i1... g ik ) D i 1...in g i1... g in ) 2 dω Ω k=0 D i 1...i n g i1... g in span{y nm (g), m = n,..., n}

Regularization D N (g)= N 3 l=0 i 1 =1... 3 i N =1 Di 1...i l g i1...g il D t (g)=e t g D(g)= N 3 l=0 i 1 =1... 3 i N =1 Di 1...i l(t) g i1...g il D i 1...i l (t) = e tl(l+1) D i 1...i l

MRI and Diffusion Tensor Imaging

Orientation Distribution Function P(R) = S(q) exp ( 2πiq R)dq ODF(g) = 0 P(r g)dr

Orientation Distribution Function: Q-ball Approximation by Funk-Radon transform ODF (g) ζ[s](g) = δ(g T w)s(w)dw w =1 w =1 δ(g T w)y lm (w)dw = 2πP l (0)Y lm (g)

Orientation Distribution Function: Q-ball 1. Fit Nth order tensor decomposition to the signal by solving linear system of equations. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = 2π( 1)k/2 (k 1)!! k!! e k(k+1)t D i 1...i k.

Diffusion Orientation transform (DOT) P(R) = S(q) exp ( 2πiq R)dq e 2πiq R = 4π P(R 0 g) = l l=0 m= l l l=0 m= l ( i) l j l (2πqr)Y lm (u)y lm(g) ( i) l Y lm (g) Y lm (u)i l(u)du

Diffusion Orientation transform (DOT)

Diffusion Orientation transform (DOT) Quality depends on R 0 There is no way to indicate optimal choice of R 0 Not robust to noise

P(R 0 g) = l,m ( i) l Y lm (g) Y lm (u)i l(u, R 0 )du I l (u, R 0 ) = 4π ODF(g) = ( i) l Y lm (g) l,m 0 J 1 (2πqR 0 )e 4π2 q 2 td(u) dq Y ( ) lm (u) I l (u, R 0 )dr 0 du 0

P(R 0 g) = l,m ( i) l Y lm (g) Y lm (u)i l(u, R 0 )du I l (u, R 0 ) = 4π ODF (g) = ( i) l Y lm (g) l,m 0 J 1 (2πqR 0 )e 4π2 q 2 td(u) dq Y ( ) lm (u) I l (u, R 0 )dr 0 du 0

R l l+3 0Γ( 2 I l (u, R 0 ) = ) 2 l+3 π 3/2 (D(u)t) (l+3)/2 Γ(l + 3/2) 1 F 1 ( l + 3 2 ; l + 3 2 ; R2 0 4D(u)t ) 1 1F 1 (a, b, z) = t a 1 (1 t) b a 1 e zt dt 0 where 0 I l (u, R 0 )dr 0 = 1 Ĩ l D(u)t Ĩ l = Γ( l+1 2 ) 4lπ 3/2 Γ(l/2)

1 D(u)t = l,m α lm Y lm (u) ODF(g) = l,m ( 1) l/2 Ĩ l α lm Y lm (g)

1 1. Fit Nth order tensor decomposition to D(u)t. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = ( 1)k/2 Γ( k+1 2 ) 4kπ 3/2 Γ(k/2) e k(k+1)t D i 1...i k.

Hardware phantom

Synthetic data Multi-tensor model n S(g) = p k e bgt D k g k=1

Q-ball and DOT-ODF comparison S(g) = S 0 e bd(g) Squared difference 9 1e 7 ODF-based method validation, without numerical 8 7 6 5 4 3 2 1 0 1000 1500 2000 2500 3000 3500 4000 b-value

Tracking Riemannian metric Finsler metric.

Finsler Metric Riemannian metric Finsler metric.

Open Questions Voxel classification Scales selection Reality check